A new approach to a non-local density functional for the calculation of electron correlation energies

Physica B 172 (1991) North-Holland

27-30

A new approach to a non-local density functional calculation of electron correlation energies

for the

C. Sommers”, S. Doniachb and C. Fub “L‘aboratoire de Physique des Solides. llniversit6 de Paris-Sud. Centre d’Orsay, F-Y1405 Orsay. France hDepartment of Applied Physics , Stanford University, Stanford, CA 9430.5, USA

The transition-metal oxides show a range of ground states of broken symmetry; for example, Mott insulators, antiferromagnetic states, charge density waves, and, more recently, high-temperature superconductors. All these properties depend essentially on electronic correlation effects induced by the Pauli effect acting on the double occupation of specific orbitals. Present first principles band theory methods using the local density approximation (LDA) do not converge to an antiferromagnetic ground state for La,_,Sr,CuO, and YBaCuO. In this paper we propose a method based on the Gutwiller projection technique to improve the treatment of the electron correlation. This method requires the reformulation of Bloch waves into localized orbitals about each site as well as a Monte Carlo method for treating the many-body Hamiltonian. We use as input to our many-body formalism the results obtained from a conventional self-consistent ASW augmented spherical wave) band calculation.

1. Introduction

The motivating force for this approach was the need to calculate correctly an antiferromagnetic ground state for the new high-T, superconductors within a one-electron picture of the solid. There is much experimental evidence for the existence of short-range antiferromagnetic order, even in the superconducting state for La,_,SrXCuO, and YBaCuO (NMR, neutrons, Mdssbauer). Even if not all physicists agree on the mechanism of high-T, superconductivity, this evidence indicates the importance of antiferromagnetic order on the ground state. Thus, a method capable of converging to this ground state should be of fundamental importance to the qualitative understanding of the high-T, superconductors. Our basic idea is to combine the local density approximation (LDA) with a configurationinteraction (CI) calculation to produce an improved estimate of the charge and spin densities p(x), (T(X), and the total energy for a crystalline solid in three dimensions. We use the wave functions resulting from a self-consistent band structure calculation as the starting values for a variational quantum Monte Carlo calculation. This calculation, based upon Gutzwiller-type trial wave functions, allows us to consider wave functions with correlations, which are pivotal in obtaining properties such as antiferromagnetism. The Monte Carlo calculation can directly produce measurements of the spin density, which have clear correspondences to experimental data, and which can potentially measure an improved electron charge density. This improved charge density can then be used with the LDA to rapidly calculate more complex properties of the material. 2. Basic formalism We extend the augmented wave functions of the form 0921.4526/91/$03.50

0

1991 - Elsevier

spherical wave (ASW) calculation

Science

Publishers

B.V. (North-Holland)

[l] beyond the LDA by considering

Yqx,, where

, .r,j) =f(.t,.

. _\,,) det(4’,(.r,))

the $,, are conventional

augmented

.

I I !

spherical

waves.

and ,f’ is

and where 1: is the variational parameter and II<,, indicates the number Thus. we must transform the wave functions to ;I site representation $,(r) =

;I

C;utzwiller

projector

crpcr-ator

of particles of spin u at Gte i. of the form

x A,,b,(r) .

( .: t

where the 4, are each well localized at site j. Our ASW basis states are centered at sites, but have long tails. However, Anderscn and Jepsen [2] pointed out that by a unitary transformation one can drastically reduce the range of the basis states while maintaining the same Hilbert space. We wish to note here that we considered using Wannier functions [6] but because of the band crossings the work involved in constructing them (because of their non-uniqueness) would tend to further complicate a11 already complicated problem. With this localized representation, the many-particle wave function has the form U(r,,

. r,,) =

det(xAl,&,(r,))

C-4)

Because of the localized nature of the 4, WC can rewrite expressions such as the cncrgy as sums over sites instead of integrals over the unit cell, computing the various two-particle integrals in advance of the However, even the number of site configurations is still extremely large, and so the evaluation sum over configurations is performed by a Metropolis algorithm. In ;I Metropolis algorithm one evaluates integrals of the form

Q

=

+IdR’WR)@VR) =

dR

&W)

P(R)

] dR !P*(R)!P(R)

WR)

(5)

'

by performing a random walk over R-space where determined by the amplitude of the configuration.

the waiting

time

at a given

point

in R-space

15

3. Localization Naively, the electrons with band index i is d(k. i) =

c A,&

are already

localized

h”(r~ R,) e”“, i ,

in the ASW

picture,

as the Bloch

wave function

at X

(6 t

where j is a site and angular momentum index within the unit cell and the 1; are the augmented Hank4 functions. However, the Hankel function only falls off inversely with distance, which is too delocalized. Anderson and Jepsen [2] pointed out that instead of using a single multipole (or in our case Hankcl function). one can screen the central function by adding functions centered at neighboring sites and

C. Sommers et al. I New approuch to LDA for electron correlation energies

29

obtain a much faster decay. Thus a typical basis function is

il,,j,R(r)= 2

c:;;:R,x,‘,,‘(I

- R’)

L',j',R'

Note that we have not fundamentally changed our Hilbert space, and thus the original ASW wave functions can be precisely represented. Anderson [2] has shown that by choosing the Cs to make the basis elements approximately orthogonal results in fairly short-ranged orbitals. To understand this we note that in the ASW method we span the unit cell with a set of overlapping spheres. The wave equation is then solved in each sphere by matching the solutions at the boundary with the spherical wave, resulting in an augmented spherical wave. To find the boundary conditions in a sphere that is not centered at the spherical wave of interest we can expand h L,

where the j, are the spherical Bessel functions. After our localization transformation, we must expand in combinations of jL and h,, as we now have Hankel functions centered on other sites. We choose to expand in

j”,-i;-cY,i,, where

This forces the overlap between two ASW to be zero to second order in the new Bs if we force the new basis states to be of the form i

L,j,R = h"L.,.R +

(11)

c B~,~:L~,~~~~~~,,~ > L'j'R'

and implies values for the Cs defined in (7). We then express a set of solutions produced by the ASW calculation, and the Hamiltonian* of the hs, that is

in terms

(13)

H one

(14)

particle

We can think of a set of Cts as corresponding

*The Madelung

potential

term here is to account

for electrons

to a coordinate

and nuclei

outside

vector R. Rewriting

the supercell

we are considering.

30

gives

t lh)

(RI). algorithm. After minimizing and we can simple R-space using a Metropolis-type to g. we can then calculate the electron charge density using the same Monte

the energy with respect Carlo method.

4. Conclusions We have begun to program our method but do not as yet have any available numerical results. Our estimates of what we cxpcct are based on calculations performed using a two-dimensional Hubbard model in conjunction with a variational Monte Carlo technique for a square lattice [IO]. This calculation has shown that the most stable phase for very small doping (3%~) consisted of both antiferromagnetism and superconductivity. Of course. these calculations do not use realistic wave functions as we propose. On the other hand. they do give us an estimate of the computing time required in the Monte Carlo parl of the calculation. If our estimate of the error made in using the LDA is correct we should not be so far away from the antiferromagnetic ground state. That is to say the corrections WC suggest should bc 01 first order. Thus one should be able to start by choosing the bands (orbitals) lying near the Fermi surface. the remaining bands and energies being calculated by ASW. [3-51. so Current limitations of computers allow us to consider systems of up to 200-300 electrons we can consider systems of about X-12 unit cells for a typical high-T superconductor. This means that we can include the highest occupied copper d and oxygen p orbitals (bands) in our calculation.

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